Validation of clustering results through correlation maps How can I compare correlation maps independently to the number of clusters in terms of  measuring the 'quality' of well separating (uncorrelated) clusters, i.e. a criterion to maximize the intra-cluster agreement and the inter-cluster disagreement?    
 A: I assume that with correlation map you mean a matrix graph similar to this but then with actual correlation values displayed as well. I also assume that by cluster you mean the result of a regression or classification tree/algorithm. If you are asking how to maximize the intra-cluster agreement and the inter-cluster disagreement, that is what a pruned regression tree would do for you automatically with a penalty for complexity (without the penalty every observation would have its own terminal node). This is usually done by minimizing a cost function like $\Sigma{RSS_k}+\lambda|k|$ where there are $k$ terminal nodes. Have a look at libraries like rpart in R.
Here I assume you are asking how to quantify the difference between the result from your clusters in a regression tree and the correlation matrix you have. I do not know if there is a formal answer for that but here is an idea:
You can combine the concept of a Gini index with Euclidean distance to give you a measure that should be comparable to the observed correlation. Whether you are validating cluster results through correlation maps or vice versa is a matter of interpretation (if you are using pearson correlation it will also give you a measure of non-linear covariance in the data).
Let us say you are looking at $m$ correlated series of data for which you ran a regression tree. Take every terminal node $k$ and determine what proportion of series $i$ is in the node. Call it $p_{ik}$. Determine this for every $i$ and you will get a vector $v_i$ of length $k$ whose sum will equal 1. Get the $\frac{m(m-1)}{2}$ euclidean distances between the $m$ vectors $v_i$ and they should be comparable with the $\frac{m(m-1)}{2}$ correlations you have. As the Euclidean distance goes to zero, the correlation should go to 1 (that is not a proven fact and probably only true in an approximate sense).
